logparameterisering

Logparameterisering (log-parameterization) refers to the practice of expressing model parameters in terms of their logarithms. This technique is used to enforce non-negativity, stabilize numerical optimization, and linearize multiplicative relationships by transforming them into additive ones. In a typical setting, a parameter θ > 0 is represented as θ = exp(φ), where φ is a free parameter that can take any real value in the estimation process.

Mathematically, if a model depends on θ, substituting θ = exp(φ) yields a log-space formulation. The derivative with respect to φ relates to the derivative with respect to θ via dL/dφ = (dL/dθ) · θ, incorporating the Jacobian of the transformation. This transformation preserves the model's predictions but changes the parameter space, which can improve conditioning and balance scale differences across parameters.

Applications include parameter estimation in statistics and econometrics, Bayesian inference with positive-constrained parameters such as rates, variances, and concentration parameters, and nonlinear models in science and engineering. In optimization, logparameterisering converts multiplicative errors and ratios into additive ones, which can simplify gradient-based methods and reduce numerical instability.

Potential drawbacks include the need to account for the Jacobian when computing priors or likelihoods in a Bayesian setting, possible changes in MCMC mixing behavior, and the introduction of nonlinearity that can complicate interpretation. The choice to parameterize in log-space should consider problem scale, prior knowledge, and the computational framework.